To find the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], [tex]\(D\)[/tex], and [tex]\(E\)[/tex] that correspond to the given circle, let's consider the general form of the equation of a circle.
The general form of the equation of a circle is:
[tex]\[ A x^2 + B y^2 + C x + D y + E = 0 \][/tex]
where [tex]\( A = B \neq 0 \)[/tex].
Since the circle has a radius of 3 units and its center lies on the y-axis, let's first write the standard form of the circle's equation:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Given:
- Radius ([tex]\( r \)[/tex]) = 3
- Center is on the y-axis, so [tex]\( h = 0 \)[/tex] and [tex]\( k = k \)[/tex].
Thus, the equation becomes:
[tex]\[ x^2 + (y - k)^2 = 9 \][/tex]
Expanding and rearranging this equation into the general form:
[tex]\[ x^2 + y^2 - 2ky + k^2 - 9 = 0 \][/tex]
Comparing this with the general form [tex]\( A x^2 + B y^2 + C x + D y + E = 0 \)[/tex]:
- [tex]\( A = 1 \)[/tex]
- [tex]\( B = 1 \)[/tex]
- [tex]\( C = 0 \)[/tex]
- [tex]\( D = -2k \)[/tex]
- [tex]\( E = k^2 - 9 \)[/tex]
Given the choices, we now need to match the values of [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], [tex]\(D\)[/tex], and [tex]\(E\)[/tex] from the choices to our values.
Looking closely at the options:
- Option A: [tex]\( A=0, B=0, C=2, D=2 \)[/tex], and [tex]\( E=3 \)[/tex] (This does not fit our form as [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are not equal to 1)
- Option B: [tex]\( A=1, B=1, C=8, D=0 \)[/tex], and [tex]\( E=9 \)[/tex] (This does not fit our form as [tex]\(C\)[/tex] is not 0 and [tex]\(D\)[/tex] is not [tex]\(-2k\)[/tex])
- Option C: [tex]\( A=1, B=1, C=0, D=-8 \)[/tex], and [tex]\( E=7 \)[/tex] (This matches our form if [tex]\( k = 4 \)[/tex], because [tex]\(-2k = -8\)[/tex] and [tex]\( k^2 - 9 = 16 - 9 = 7 \)[/tex])
- Option D: [tex]\( A=1, B=1, C=-8, D=0 \)[/tex], and [tex]\( E=0 \)[/tex] (This does not fit our form as [tex]\(C\)[/tex] is not 0 and [tex]\(D\)[/tex] is not [tex]\(-2k\)[/tex])
- Option E: [tex]\( A=1, B=1, C=8, D=8 \)[/tex], and [tex]\( E=3 \)[/tex] (This does not fit our form as [tex]\(C\)[/tex] is not 0 and [tex]\(D\)[/tex] is not [tex]\(-2k\)[/tex])
Thus, the set of values that correspond to the equation of the circle is:
[tex]\[ \boxed{C. \ A=1, B=1, C=0, D=-8, E=7} \][/tex]
